A swiss cheese theorem for linear operators with two invariant subspaces
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- by Audrey Moore and Markus Schmidmeier PDF
- Proc. Amer. Math. Soc. 143 (2015), 5097-5111 Request permission
Abstract:
We study systems $(V,T,U_1,U_2)$ consisting of a finite dimensional vector space $V$, a nilpotent $k$-linear operator $T:V\to V$ and two $T$-invariant subspaces $U_1\subset U_2\subset V$. Let $\mathcal S(n)$ be the category of such systems where the operator $T$ acts with nilpotency index at most $n$. We determine the dimension types $(\dim U_1, \dim U_2/U_1, \dim V/U_2)$ of indecomposable systems in $\mathcal S(n)$ for $n\leq 4$. It turns out that in the case where $n=4$ there are infinitely many such triples $(x,y,z)$, they all lie in the cylinder given by $|x-y|,|y-z|$, $|z-x|\leq 4$. But not each dimension type in the cylinder can be realized by an indecomposable system. In particular, there are holes in the cylinder. Namely, no triple in $(x,y,z)\in (3,1,3)+\mathbb N(2,2,2)$ can be realized, while each neighbor $(x\pm 1,y,z), (x,y\pm 1,z),(x,y,z\pm 1)$ can. Compare this with Bongartz’ No-Gap Theorem, which states that for an associative algebra $A$ over an algebraically closed field, there is no gap in the lengths of the indecomposable $A$-modules of finite dimension.References
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Additional Information
- Audrey Moore
- Affiliation: Department of Mathematical Sciences, Delaware State University, 1200 N DuPont Highway, Dover, Delaware 19901
- Address at time of publication: Department of Mathematical Sciences, Florida Atlantic University, 777 Glades Road, Boca Raton, Florida 33431
- MR Author ID: 915495
- Email: audreydoughty@yahoo.com
- Markus Schmidmeier
- Affiliation: Department of Mathematical Sciences, Florida Atlantic University, 777 Glades Road, Boca Raton, Florida 33431
- MR Author ID: 618925
- ORCID: 0000-0003-3365-6666
- Email: markus@math.fau.edu
- Received by editor(s): September 22, 2014
- Published electronically: May 20, 2015
- Additional Notes: This research was partially supported by a Travel and Collaboration Grant from the Simons Foundation (Grant number 245848 to the second-named author).
- Communicated by: Harm Derksen
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 143 (2015), 5097-5111
- MSC (2010): Primary 16G20, 47A15
- DOI: https://doi.org/10.1090/proc12754
- MathSciNet review: 3411129